You're a farmer, and are moving a sheep dog, a chicken, and a bag of feed across a river to your new farm. Being so poor, you can only afford a boat which can carry yourself and one other object. Your animals are very hungry; if you leave the sheep dog and the chicken alone, the dog will eat the chicken. If you leave the chicken and feed alone, the chicken will eat the feed. How do you get all three across safely?

There's a row of 100 lightbulbs, numbered 1-100, and 100 frogs, also numbered. Whenever a frog jumps on a lightbulb, it toggles the bulb on/off. Frog n will jump on every nth bulb (example, the 4th frog will jump on the 4th, 8th, 12th, 16th, etc. bulbs). When all the frogs are done, which bulbs are on?

You have 3 switches in the basement, labeled A, B, and C. In the attic are 3 lightswitches, 1, 2, and 3. Each switch corresponds to one lightbulb (ie, A might turn on 3, B = 2, and C =1). You can easily figure out which turns on which with 3 trips to the attic. If you're clever, you can do this in two trips. You, however, want to do it in one trip. Can you?

You have three boxes, each with two marbles. There are 2 marble types: black and white. Each box contains two marbles; one box both white, one both black, and one white and one black. Each box is labeled based on its contents, BB, WB, and WW; someone, though, has come back and changed the labels so that all the boxes are mislabeled. You are allowed to take one marble out of a box at a time, without looking inside. What is the leastnumber of marbles you must remove to figure out which box is which?

Part I: You work the nightshift as a securityguard, and, while getting ready for work, the power keeps going out. You know that in your sock drawer are 10 white socks and 10 black socks (you're too tired to fold them). How many socks do you need to draw from the drawer to ensure you have a pair of socks?

Part II: New companypolicy: black socks only! How many socks do you need to draw from the drawer to ensure you have a pair of black socks?

Four people are standing in a row facing the same direction, to the right. The person in the back of the line is asked to turn away, to look left. They are all asked to close their eyes while a hat is put on their head randomly. The hats are drawn from a bag of 2 white and 2 black hats. Whenever someone logically deduces what color hat they have on, they get a prize. So they open their eyes ... About a minute passes, and someone says "I know what color my hat is!" Who was it and how did they figure it out?

Answers (I had to figure these out myself; if you can find the answers to those I haven't got, or if you find any errors, please send them to me.

Lightbulbs and Switches: You can! Here's how: Turn on one switch and wait 5 minutes or so, then turn it off and flick on another switch. Quickly run upstairs. The lightbulb which is on belongs to the switch you just turned on, the lightbulb which is hot is connected to the switch which you first turned on, and the cold lightbulb is connected to the remaining switch.

The Disfigured Checkerboard: Well, the corner's of the board which are removed will be of the same color. This means that there will be 32 of one color and 30 of the other. No matter what, after placing 15 dominoes, there will be two remaining squares of the same color. Since no two adjacent squares are the same color, it is impossible to do this. (Thanks to Mr. Ibarra for this answer)

Scrambled Marbles: Only 1! First, take a marble from the box labeled "BW". This box must be either BB or WW (it is mislabeled), so if the marble is black, the box should be BB; if white, WW. So you switch the signs so that the one box is labeled correctly. Now, one box is labeled with BW and one with WW/BB. The one with WW/BB is mislabeled (it hasn't changed since you started), and it can't be the label of the now correctly labeled box. This means that it has to be BW. So you switch signs again and they are now labeled correctly. Here's an example of how this works:

You draw a black from the box labeled "BW". This means that this box has 2 black marbles in it

The box labeled "WW" can't have 2 white marbles in it; we know it is labeled wrong.

The box labeled "WW" can't have 2 black marbles in it; we already know which box has 2 black marbles.

Therefore, the box labeled "WW" must have 1 black marble and 1 white marble. It should be labeled "BW"

Also, the box labeled "BW" must have 2 white marbles in it.

You switch the signs and the boxes are all correctly labeled now.

The Security Guard: 3 and 12. Part I: After 2 socks, it is possible for you to draw one black and one white sock. The third sock gives you a pair. Part II: You could conceivably draw 10 white socks in a row. So the 11th and 12th socks, if this happens, must be black.

The Tower and the Glass Balls: I'm thinking drop a ball at every third floor; if a ball breaks, go down two floors and drop your remaining ball. If it doesn't break, go up one floor and drop it there. This seems silly, though, because the number 3 seems arbitrary. This seems wrong, come up with a better answer and write me!

Hats: The person second from the front figured it out. How? Because the third person didn't give an answer right away. The only way for the third person to be able to answer right away, the two front hats would have to be the same color (then the third person would know his/hers was the opposite color). Since the third person is silent, then the front two hats must be of opposite colors. Therefore, the second person simply must look at the front person's hat, wait, then say the opposite color. This was tricky, cause it draws you into thinking about that third person who has the most informationimmediately.

Frogs and Lightbulbs:
There's an elegant way to figure this out using
number theory. The way I actually did it, however, was just to test it out and look for a pattern.
I'll list which bulbs are on after each frog hops.

1 2 3 4 5 ... 100

1 3 5 7 9 ... 99

1 5 6 7 10 12 13 17 18 ...

1 4 5 6 7 8 10 13 17 18 ...

1 4 6 7 8 13 15 17 18 ...

1 4 7 8 12 13 15 17 ...

1 4 8 12 13 14 15 17 ...

1 4 12 13 14 15 16 17 ...

1 4 9 12 13 14 15 16 17 18...

Ah ha!
1, 4, and 9 will be on at the end. 2, 3, 5, 6, 7, and
8 will not. You could keep on going; it will probably be
even more obvious after you reach 25. The square numbers
(and 1) are left on.

The elegant way: Consider a lightbulbk. Which frogs will
toggle k? All the frogs whose number is a factor of k. Since every lightbulb starts in the off position, only numbers with an odd
number of factors will be on after all frogs have hopped. Which numbers have an odd number of factors?
From number theory, only perfect squares have an odd
number of factors. Well, and the number 1, of course.

Election Logic:
This is a standard "enumerate every possibility" puzzle.
There are two liars and one truth teller. Try each possible
man as the truth teller, and look for a contradiction. Return the first guess that doesn't reach a contradiction.

Guess 1: George is telling the truth.
George's true statement -> Al didn't win.
Al's false statement -> one of Al or George won.
Ralph's false statement -> neither George nor Al won.
Contradiction: (one of Al or George won) and (neither George nor Al won) can not both be true.

Drop a ball every
square_root(n) floors until it breaks, and then try every floor sequentially from the next lowest multiple of
square_root(n) until the second ball breaks.

Worst
case here will be 2*square_root(n) trips up and down the tower, and O(square_root(n)) trips in the average case.
MinderBender's solution was n/3 and O(n/3) for the
worst case and average case, respectively. This
seems pretty elegant, but I haven't proved that it's the
most efficient. I've also left out the caveat about having to go up and down the tower only every two times when you still have both balls, but this will just be something like a factor of (2/3) (square_root solution) or
(1/2) (3 floors solution), and does not effect the asymptoticbehavior.